Study design

A cross-sectional study was conducted to examine geographical, religious, and residential factors associated with maternal age at first birth (AFB) in Ethiopia.

Study setting and population

Ethiopia is located in the Horn of Africa and is bordered by Eritrea to the north, Djibouti and Somalia to the east, Kenya to the south, South Sudan to the west, and Sudan to the northwest. Administratively, the country is divided into nine regional states Tigray, Afar, Amhara, Oromia, Somali, Benishangul-Gumuz, Southern Nations, Nationalities and Peoples’ Region (SNNPR), Gambela, and Harari and two chartered cities, Addis Ababa and Dire Dawa.

This study utilized secondary data from the 2019 Ethiopia Mini Demographic and Health Survey, which was conducted from March 21 to June 28, 2019. The survey was implemented by the Ethiopian Public Health Institute in collaboration with the Central Statistical Agency (Ethiopia) and the Federal Ministry of Health (Ethiopia), with technical assistance from ICF. The Mini EDHS is a nationally representative household survey designed to provide reliable estimates of demographic and health indicators at the national and regional levels, as well as by place of residence (urban and rural). The dataset used in this study was obtained from the Demographic and Health Surveys Program website after formal approval of a data access request. Ethical clearance for the survey was obtained by the DHS Program from the Institutional Review Board of ICF and the national ethics review committee in Ethiopia [32].

The target population of the Mini EDHS includes women aged 15–49 years residing in selected households across Ethiopia. However, for the purposes of this analysis, the study population was restricted to women aged 15–49 years who had experienced at least one birth before the survey date.

Sample design

The 2019 Ethiopia Mini Demographic and Health Survey employed a two-stage stratified cluster sampling design. In the first stage, 305 census enumeration areas (EAs) were selected from the national sampling frame, including 93 urban clusters and 212 rural clusters. In the second stage, a systematic sampling method was used to select approximately 30 households from each selected enumeration area, resulting in a total sample of 8,885 households. The sampling design ensured adequate representation across regions and urban–rural areas, with proportional allocation applied to most regions and additional sampling consideration for the three largest regions: Amhara, Oromia, and SNNPR. This design allowed the survey to produce nationally and regionally representative estimates of key demographic and health indicators.

Analytical sample

Age at first birth is inherently a time-to-event variable, and women who have not yet experienced a first birth represent right-censored observations. In the present study, the analysis was intentionally restricted to women who had already experienced their first birth by the time of the survey. This restriction aligns with the study objective of modeling the timing of first birth among women who have transitioned to motherhood and allows direct estimation of the conditional mean and dispersion of maternal age at first birth.

Including women who had not yet experienced a first birth would require survival analysis methods to account for censoring, which was beyond the scope of the present modeling framework. Consequently, the analytical sample consisted of 5,839 women aged 15–49 years who reported having had at least one live birth at the time of the survey. Women who had not yet experienced a first birth were excluded from the analysis.

While this restriction allows detailed modeling of the distribution of age at first birth among mothers, it changes the interpretation of results. Therefore, the findings should be interpreted as applying to women who had already experienced first birth at the time of the survey, rather than to all reproductive-age women in Ethiopia.

Outcome variables

The study outcome variable is maternal age at first child birth.

Explanatory variables

Region (Tigray, Afar, Amhara, Oromia, Somali, Benishangul, SNNPR, Gambela,Harari, AddisAbaba and Dire Dawa).

Place of residence (Urban, Rural).

Religion (Muslim, Orthodox, Protestant, Catholic, Traditional, Others).

The selection of explanatory variables in this study was guided by considerations of temporal consistency with the outcome variable, maternal age at first birth (AFB). Although maternal education, occupation, and marital status are widely recognized in the literature as important correlates of AFB, these variables in the 2019 Ethiopia Mini Demographic and Health Survey were recorded at the time of the survey rather than at the time when the first birth occurred for most respondents. Because these characteristics may have changed substantially after the first birth event, their inclusion could introduce temporal inconsistency and potential reverse causality. For example, early childbearing may itself influence subsequent educational attainment or employment status, thereby biasing the estimated relationships if these variables were included as predictors.

Nevertheless, the exclusion of such variables may introduce omitted variable bias if they are associated with both the outcome and the retained covariates. For instance, educational attainment is likely correlated with regional differences in access to schooling as well as with delayed childbearing. Consequently, some of the estimated regional differences in maternal age at first birth may partly reflect underlying educational disparities. Similarly, marital status and occupation may be linked to both social norms and reproductive behavior, and their exclusion could lead to over- or underestimation of the associations between the included covariates and age at first birth. These limitations are acknowledged, and the results should therefore be interpreted as associations rather than causal effects.

Given these considerations, the final set of explanatory variables included region, religion, and place of residence. These variables were selected because they represent relatively stable contextual characteristics that are less likely to change following the occurrence of first birth. They capture important geographic, cultural, and social dimensions that may influence reproductive behavior, including differences in access to education and health services, sociocultural norms surrounding marriage and childbearing, and disparities between urban and rural environments. Focusing on these contextual variables allows the analysis to explore patterns and associations in maternal age at first birth among women who had already experienced a first birth at the time of the survey, while minimizing temporal inconsistencies between predictors and the outcome.

Ethical approval and consent

This study utilized publicly available, de-identified data from the 2019 Ethiopia Mini Demographic and Health Survey (Mini EDHS), obtained with permission from the DHS Program (www.dhsprogram.com). Ethical approval for this secondary analysis was waived by the Kotebe University of Education Institutional Review Board by letter with ref No Math098/2017, as the data are anonymized and publicly accessible, preventing identification of individual participants. The study adhered to ethical standards for secondary data analysis, with no issues related to dual publication, misconduct, or participant confidentiality.

Data analysis

The variable of interest in this research is the age at which a mother gives birth for the first time. This is calculated as the number of years from her date of birth to the age at which she delivers her first child. The dataset on age at first birth is considered to be positively skewed and continuous, as it consists of a series of positive values within limited range. When dealing with such data that is continuously distributed, positively skewed, and always positive, one common technique is to perform a log transformation on the data [33]. However, there are other approaches within the generalized linear model framework that may provide a more comprehensive analysis of this type of dataset. It is important to explore these alternative methods to ensure that the analysis of the age at first birth data is both accurate and meaningful. In this study, it was observed that the dataset on the maternal age at first birth obtained from the Mini EDHS 2019 dataset (excluding mothers without birth during survey period) follows a gamma distribution and consists of only positive values.

Gamma regression was chosen because the focus of this study is on modeling the observed age at first birth as a continuous, positively skewed variable, rather than modeling the hazard or timing of the event. The Gamma distribution naturally accommodates skewness and strictly positive values, allowing both the mean and dispersion to be modeled flexibly under GLM, GLMM and distributional Gamma regression frameworks. In contrast, time-to-event models such as the Cox proportional hazards or Accelerated Failure Time (AFT) models are most appropriate when dealing with right censored data or when the objective is to model the hazard rate over time. In this study, all respondents had experienced the event (no right-censoring), and the research objective was to quantify the relationship between regional, religious and residential covariates and the mean age at first birth and variability, making Gamma regression a more direct and interpretable choice.

This study explores Gamma regression models that allow both the mean and shape (dispersion) parameters to depend on covariates. Initially, the mean parameter is modeled within the frameworks of the generalized linear model (GLM) and generalized linear mixed model (GLMM). Subsequently, a distributional Gamma regression model is employed, in which both the mean and variability are regressed on relevant covariates. All models are examined under both classical (maximum likelihood) and Bayesian (MCMC) estimation approaches [33].

Modeling framework and estimation approaches

The Linear Model (LM) has had a very broad development. LM has developed into the Generalized Linear Model (GLM). Furthermore, the combination of GLM and Linear Mixed Model (LMM) can form a Generalized Linear Mixed Model (GLMM) [34]. The generalized linear model (GLM) is a powerful regression model that merges linear and nonlinear elements, enabling the incorporation of non-normal response distributions. In a GLM, the response variable needs to be a part of the exponential family, which encompasses various familiar distributions like normal, log normal, binomial, exponential, Weibull and gamma distributions. The normal-error linear model is a specific variant of the GLM, making the GLM a versatile tool for empirical modeling and data analysis [34]. Explanatory variables in GLMs can be either continuous or categorical. The GLM determines a separate regression coefficient for each level of categorical variables indicating the need for exercising considerable care in interpreting that coefficient.

The GLM is constructed from three key components:

1. Random Component: This specifies that the conditional distribution of the response variable (the i^th of n independently sampled observations) is determined by the values of the explanatory variables, with the expected value E () = for i = 1,...,n. It is assumed that follows a specific probability distribution, such as the gamma distribution. The choice of distribution is contingent upon the type of response variable and the characteristics of the data [35].
2. Systematic Component: This defines a linear predictor, which is a linear combination of the explanatory variables with k predictors, represented as . This is expressed as a linear predictor , where the relationship can be formulated as where being the vector of unknown regression coefficients that need to be estimated [35].
3. Link Function: This component connects the linear predictor to the mean of . The link function is a monotonically increasing function, and the selection of this function depends on the distribution of . For instance, when represents a positive integer, the mean E () = must also be non-negative, prompting the use of a log link function, which expresses the mean in terms of the entire real line. In this case, the link function is represented as log effectively relating to the linear predictors. Consequently, log-linearity for the mean is frequently employed as the link function within the regression model.

Generalized linear mixed models (GLMMs) is an extension of GLM where random effects are included in the model. GLMMs are a natural outgrowth of both linear mixed models and generalized linear models. GLMMs can be developed for non-normally distributed responses, will allow nonlinear links between the mean of the response and the predictors, and can model over dispersion and associate by incorporating random effects (Breslow NE, Clayton DG. 1993).

In both generalized linear models (GLMs) and generalized linear mixed models (GLMMs), the emphasis is typically on estimating the mean structure, assuming that dispersion remains constant. However, real-world data can sometimes show more variability than what is predicted by the standard mean-variance relationship. When dispersion actually varies, relying on a constant dispersion assumption can significantly reduce the efficiency of mean parameter estimates [36]. To address this, a more flexible approach involves jointly modeling both the mean and the dispersion, allowing dispersion to be expressed as a function of covariates [37].

This study addressed the facts related with the trends, factors and model performances by focusing on the response variable, which is the maternal age at first birth in Ethiopia. Given the nature of the data, Gamma distribution was used for analysis. By leveraging the GLM, GLMM and distributional Gamma regression modelling framework with a log link function, it became feasible to establish a linear relationship between the predictors and the response variable. The log link function is used in Gamma regression because it ensures that predicted values remain strictly positive, aligning with the domain of the Gamma distribution. It transforms the nonlinear relationship between predictors and the response variable into a linear one on the log scale, facilitating interpretation and estimation. This transformation allows model coefficients to be interpreted multiplicatively, meaning a unit change in a predictor corresponds to a percentage change in the response. Additionally, the log link accommodates the Gamma distribution’s variance structure, where variance increases with the square of the mean, and it enhances numerical stability during model fitting by preventing negative predictions and improving convergence. This approach, allows for analysis and interpretation of the factors influencing maternal age at first birth in the Ethiopian context.

Gamma regression

The gamma distribution is commonly used to represent continuous data that is restricted to positive values. When a variable (𝑦) follows a gamma distribution (𝑦, 𝛼, 𝛽), its probability density function is determined by its mean 𝜇 (𝜇 > 0), shape parameter 𝛼 (𝛼 > 0), and rate parameters 𝛽 (𝛽 > 0). The probability density function for the gamma distribution is outlined by [37] as follows:

(1)

Where:

𝑦: Dependent variable.

Γ (𝛼): Gamma function is defined as

Let 𝛼 = 𝜈 and 𝛽 = 𝜈/𝜇 ⇒ 𝜇 = 𝜈/𝛽

(2)

Then 𝑦 ~ 𝐺 (𝜈,). In this case E(y) = and var(y) =

If (𝑦) has density function (2), we write 𝑦 ~ (𝜇, 𝜈) [37]. Then the gamma regression model is defined as follows:

(3)

where 𝜖 is vector of random error with a dimension (n × 1), and (n) is the sample size.

One way to transform the gamma regression function to a linear form is to use the logarithm of the mean () as the response variable. This can be achieved by taking the natural logarithm of both sides of the equation (3) [38].

In order to analyze the determinants influencing the maternal age at first birth, three models were considered in GLM framework: The generalized linear model (GLM) (model 1) presented in equation (4), generalized linear mixed model (GLMM) (model 2), being in the presence of random effects and finally distributional Gamma regression model (model 3).

The GLM (model 1) generalizes a linear model and is composed of two fundamental elements. The first is the link function between the expected value of the maternal age at first birth and the linear predictors, and the second one is the variance function, in which the variance is expressed as a function of the mean. To explain the structure of the GLM, the linear model starts from:

(4)

where is the 𝑖^th row of X = [, , …, ]′ of X which is an (n × p) data matrix with (p) explanatory variables, β is a (p  × 1) vector of the slope coefficients and = g() is the log link function for the gamma regression model [38].

The second model applied was the GLMM (model 2) presented in equation (5). The GLMM is an extension of the GLM. The GLM includes in the mathematical model only the fixed effects, estimating their influence on the dependent variable. Instead, the GLMM considers, in addition to the fixed effects, the random effects. The random effects of the GLMM were the individual mothers in different enumeration areas, since the variability of maternal age was depend on the intrinsic characteristics of the mothers. In the gamma regression model using GLMM framework presented by [38], the shape parameter is constant for every observation, and it is modeled like the mean.

Here, we assume an independent random variables, ; i = 1,…….., n, with mean parameter given by

(5)

where is fixed effect coefficients and , are the sets of random effect coefficients, and are observed values of the fixed and random covariates respectively, and is the linear predictors at the i^th observation. Again, it is convenient to take the first component of the X_i vector as 1 to allow for an intercept in the mean regression structure.

Sometimes modeling each parameters of the distribution parameters with covariates can be more informative than the GLM and GLMM application [39]. Using distributional Gamma regression instead of standard GLM (Generalized Linear Model) or even GLMM (Generalized Linear Mixed Model) is beneficial when both the mean and dispersion (variability) of the outcome are systematically influenced by covariates.

In the classical parametric regression, both the mean and shape (inverse of dispersion) are modeled as functions of covariates via log-link functions as expressed in equation (6) and (7).

(6)

(7)

is a vector of covariates (e.g., region, religion, residence) for observation β is the vector of regression coefficients for the mean, is a vector of covariates (may be same or different from ) for the shape, and γ is the vector of regression coefficients for the shape. Parameter estimates β and γ are obtained via Maximum Likelihood Estimation (MLE) and Bayesian method.

Maximum likelihood estimation

Adekanmbi V. (2017) [38] propose a classic approach to fit gamma regression model, where mean parameter follow regression structures in constant shape parameter, using the Fisher scoring algorithm. In that work, it was shown that for gamma re-parameterization given by (2) the GLM likelihood function can be written in the form:

(8)

and the log-likelihood function by:

(9)

Thus, assuming the regression structures defined by the score statistics are given by:

(10)

and the Hessian matrix is determined by:

(11)

In matrix form

Thus, the Fisher information matrix is given by:

(12)

Similarly for the case of Model 2: Gamma GLMM (Fixed plus Random Effects), when the liner predictor become and random effect b ~ N (0, D) the conditional likelihood become:

(13)

Log-likelihood function become

(14)

The Hessian matrix become

(15)

Where

It can be noted that the Fisher information matrix is a block diagonal matrix, where one of the blocks corresponds to the mean regression parameters and the other to the random effect b. Thus, and are orthogonal and their maximum likelihood estimators and are asymptotically independent. As a consequence of this result, (Adekanmbi V., 2017) [38] proposed an iterative algorithm to obtain the maximum likelihood estimates of the regression parameters.

In the similar way the distributional gamma regression model (model 3) the likelihood function is derived as follows

(16)

Where and are the location and shape (inverse of dispersion) respectively and those expressed as regression coefficient as follows

The log-likelihood for a sample of n independent observations is

(17)

Finally we can derive the score function (first derivatives) and the Hessian matrix (second derivatives) and fisher information for parameters and we assume , , and .

Where Ψ(ϕ_i) is digamma function (the first derivatives of )

The Hessian matrix become

is trigamma function (the first derivatives of digamma function)

After developing such mathematical derivation we can estimate the values of the parameters vector and using iterative optimization in Newton-Raphson algorithm.

Bayesian estimation

Combining the prior probability density function of the parameters with the likelihood function of the observations yields the Bayesian estimator, which is only the information-rich part of the posterior probability density function [40]. The Bayesian method to fit gamma regression models, where mean parameter follow regression structure. As in these works, to implement a Bayesian approach to estimate the parameters of the gamma regression model, we need to specify a prior distribution for the parameters and the data (Y) likelihood and relate to the posterior density. The relation become:

(18)

Where, is the posterior distribution of the parameters and given the data, is the likelihood of the data given the parameters, is the prior distribution of the parameters.

The first procedure to formulate the posterior distribution is specifying the likelihood function in both GLM and GLMM framework. For the case of GLM the likelihood function is formulated in Equation (6) as

(19)

and for GLMM the likelihood function is formulated in equation (11) as

(20)

Secondly, fix the prior distribution depending on the data nature or by highlighting previous literature about maternal age at first birth specifically in Ethiopia. Mathematically the prior for GLM Gamma regression model coefficients a common choice is a multivariate normal prior for the regression coefficients:

(21)

And for GLMM case the fixed prior and the random effect prior is needed

(22)

For the intercept β₀ ~N (3, 0.5) based on the data nature and previous literature, and the other fixed coefficients β~ N (0, 1).

Lastly using equations (15–18) the expressions for the likelihood and priors [41], the full posterior distribution for Gamma GLM regression can be formulated as

(23)

The extended posterior by the effect of random component become

(24)

The mathematical formulation of the Bayesian estimation for the joint modeling of mean and dispersion in gamma regression, the likelihood function is:

(25)

By assuming the normal prior for the regression coefficients

(26)

(27)

Using Bayes’ theorem, the joint posterior is expressed using equation (25), (26) and (27)

(28)

Stan Modeling: Stan is a specialized state dialect implemented as a C++ library designed for Bayesian modeling, as noted by [41]. It serves as a Bayesian programming tool that primarily employs the No-U-Turn Sampler (NUTS) to perform posterior simulations based on user-defined models and data. Hamiltonian Monte Carlo (HMC), is a method that falls under the broader category of Markov Chain Monte Carlo (MCMC) techniques. While HMC can be quite intricate due to the need for calculating potentially complex derivatives and fine-tuning several parameters, it often proves to be faster and more reliable than simpler methods like basic Markov chain simulation, Gibbs sampling, and the Metropolis algorithm, particularly in more complex scenarios, as it navigates the posterior parameter space more effectively.

They achieve this by associating each model parameter with a momentum variable, which influences the exploration behavior of Hamiltonian Monte Carlo (HMC) based on the posterior density of the currently sampled parameter. This approach allows HMC to reduce the random walk characteristics typical of the Metropolis algorithm [41]. As a result, Stan demonstrates significantly greater efficiency compared to conventional Bayesian software algorithms like Metropolis and Gibbs. Notably, the primary function of the rstan package is to interface with the Stan software to estimate a given statistical model, and rstan features an intelligent system that automates much of the adaptation process.

A statistical model can be represented through a conditional probability function P( | y, x), where () denotes a series of unknown values being modeled, y represents a series of known response variable values, and x includes un modeled predictors and constants, such as sizes and hyper parameters. A Stan program explicitly defines a log probability function for parameters based on the provided data and constants. Stan facilitates comprehensive Bayesian inference for continuous-variable models using Markov chain Monte Carlo methods, including an enhanced version of Hamiltonian Monte Carlo sampling [42]. Users can access Stan from R via the rstan package. This interfaces support sampling and optimization-based inference, along with diagnostics and posterior analysis. Additionally, rstan offer functionalities for accessing log probabilities, transforming parameters, and creating specialized plots. Stan programs are structured with variable type declarations and statements, which include constrained and unconstrained integers, scalars, vectors, and matrices. Variables are organized into blocks that correspond to their intended use: data, transformed data, parameters, transformed parameters, or generated quantities.

Model Diagnosis Checking: model 1, model 2 and distributional gamma regression were specified as in the gamma family and with log link. For the case of maximum likelihood estimation method (MLE), the goodness of fit for GLM, GLMM and distributional Gamma regression was checked by the Akaike information criterion (AIC), Bayesian information criterion (BIC) and log likelihood test; the best model is the one that minimizes these values. For the case of Bayesian technique the goodness of fit the performance of each model was evaluated by using potential scale reduction factor statistic, effective sample size value, Monte Carlo standard error (MCSE) and posterior predictive checking method. Loo method with expected log pointwise predictive density and their standard errors were used for comparison of GLM, GLMM and distributional Gamma regression.

Descriptive characteristics of the study sample

A total of 5,839 women aged 15–49 years who had experienced at least one live birth were included in this analysis using data from the 2019 Ethiopia Mini Demographic and Health Survey. Among these women, the mean maternal age at first birth was 18.32 years. Maternal age at first birth among the respondents ranged from 10 years to 40 years, indicating substantial variability in the timing of entry into motherhood among women who had already experienced first birth at the time of the survey.

Table 1 presents the geographic, residential, and religious characteristics of the study participants. The regional distribution of respondents ranged from 6.4% in Addis Ababa to 12.1% in Oromia. The largest proportions of participants were from Oromia (12.1%), SNNPR (11.8%), and Amhara (11.0%), reflecting the relatively larger populations of these regions in Ethiopia. With respect to place of residence, 71.8% of respondents lived in rural areas, while 28.2% resided in urban areas. This distribution reflects the predominantly rural composition of Ethiopia’s population. Regarding religious affiliation, Muslim women represented the largest group (42.8%), followed by Orthodox Christians (35.3%) and Protestants (19.9%).

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Table 1. Geographic, residential and religious characteristics of study participants, extracted from Mini EDHS 2019 survey dataset (Ethiopia), that conducted from March 21, 2019, to June 28, 2019 (N = 5839).

https://doi.org/10.1371/journal.pone.0346504.t001

Smaller proportions were observed for Catholics (0.6%), traditional religion followers (1.1%), and other religions (0.3%). These descriptive statistics provide an overview of the socio-demographic composition of the analytical sample consisting of women who had already experienced first birth at the time of the survey.

Fig 1 presents the histogram and density plot of maternal age at first birth among women who had experienced first birth. The distribution is positively skewed, with a concentration of observations in the late teenage years and a longer right tail extending toward older ages. As expected, the outcome variable takes only positive values, consistent with the nature of age-at-event data.

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Fig 1. Maternal Age at first birth histogram and density.

https://doi.org/10.1371/journal.pone.0346504.g001

Fig 2 illustrates boxplots of maternal age at first birth across selected covariate categories, including region and religion. The boxplots suggest variability in the distribution of maternal age at first birth across these groups. A small number of outliers are observed in several categories, reflecting cases of unusually early or late first births.

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Fig 2. Maternal age at first birth box plot on regions and religion.

https://doi.org/10.1371/journal.pone.0346504.g002

Comparison of candidate distributions

Fig 3 compares the fitted Gamma, Normal, log-normal and Weibull distributions to the observed maternal age at first birth of women who had already give birth in Ethiopia. The gamma distribution (blue curve) provides the best visual fit to the observed histogram, capturing both the central peak around the late teens and the gradual decline toward older ages. Its flexibility in modeling positively skewed, non-negative data makes it well-suited for maternal age at first birth, where the variable cannot take values below puberty and tends to have a longer right tail. The gamma fit closely follows the observed frequency in both the lower and middle age ranges, and its right-tail decay is consistent with the rarity of very late first births. This close alignment suggests that the gamma distribution’s shape and scale parameters effectively represent the underlying variability in the data.

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Fig 3. Maternal age at first birth curve (black) with gamma and normal distribution curves.

https://doi.org/10.1371/journal.pone.0346504.g003

The normal distribution (red curve) captures the general center of the distribution but performs less well in the tails. Because the normal is symmetric, it slightly overestimates the frequency of very low and very high ages and underestimates the skewness present in the data. In contrast, the log-normal distribution (green curve) show discrepancies which implies the log-normal less appropriate for this dataset compared to the gamma, despite its theoretical appeal for positively skewed variables.

The Weibull distribution (purple curve) offers a moderately good fit, particularly in the lower to mid-range of ages, but it underestimates the peak and slightly overshoots in the left tail. While the Weibull is also flexible and can accommodate skewed, non-negative data, its parameterization here does not align as closely with the observed pattern as the gamma does. This may be because the Weibull’s hazard-based shape is less suited to the gradual tail decay of maternal age at first birth, compared to the gamma’s smoother decline. The gamma emerges as the most balanced fit across the entire range of ages in this dataset.

Given these comparisons, the gamma distribution emerges as the most appropriate model for maternal age at first birth in this dataset, offering a close empirical fit while aligning with the theoretical expectation that such demographic events follow a positively skewed, non-negative distribution. Its flexibility in capturing variation across regions, religions, and residential contexts supports our conceptual framework, which emphasizes the interplay of structural and cultural factors in shaping the timing of first births. By adopting the gamma distribution, our subsequent regression analyses can more accurately reflect the underlying data structure, thereby providing more reliable estimates of the associations between key socio-demographic predictors and maternal age at first birth.

The result in Table 2 supports the evidence in Fig 4 that Gamma distribution emerged as the strongest candidate for modeling the maternal age at first birth data, as indicated by its lowest AIC (−5427.93) and BIC (−5051.38) values among the four candidate models. Its log-likelihood (−2776.97) was also the highest (least negative), suggesting superior fit to the data. The Kolmogorov–Smirnov (K–S) statistic for the Gamma model was 0.14 with a p-value of 0.292, which is above the conventional significance threshold of 0.05, indicating no evidence to reject the null hypothesis that the data follow this distribution. This combination of excellent information criteria performance and acceptable goodness-of-fit test results provides strong empirical support for the Gamma model’s suitability.

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Table 2. Model Comparison for Maternal Age at First Birth Data Using Information Criteria and Goodness-of-Fit Statistics.

https://doi.org/10.1371/journal.pone.0346504.t002

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Fig 4. Residual diagnosis.

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The Normal distribution performed less favorably, with higher AIC (−5341.22) and BIC (−4951.23) values compared to the Gamma model. While its K–S statistic (0.112) was slightly lower than Gamma’s, the associated p-value (0.034) fell below 0.05, suggesting a statistically significant departure from normality in the data. This aligns with theoretical expectations, as maternal age at first birth data are often positively skewed, a feature that the symmetric Normal distribution cannot adequately capture. Consequently, despite being a common baseline model, the Normal distribution appears unsuitable for this application. The Log-normal and Weibull distributions showed intermediate performance between Gamma and Normal. The Weibull model achieved lower AIC (−5303.09) and BIC (−5099.88) than Log-normal, and both had K–S p-values (0.041 for Log-normal, 0.050 for Weibull) near or at the significance threshold, suggesting marginal fits. While Weibull model can accommodate skewness better than the Normal distribution, it still failed to match the fit quality of the Gamma model, as reflected in their higher AIC/BIC values and slightly less favorable K–S results.

Diagnosis checking

In the maximum likelihood estimation within gamma regression, model diagnosis hinges on scrutinizing gamma regression residuals. This analytical process is geared towards pinpointing outliers and potential model misspecifications.

Assessing the distribution of residuals through a Quantile-Quantile (QQ) plot, as described in Fig 4, reveals that the residuals closely mimic a normal distribution. This alignment suggests that the residuals follow a normal distribution pattern, a favorable situation in model evaluation.

Model performance and model comparison

To evaluate the adequacy of alternative modeling frameworks for maternal age at first birth (AFB), three Gamma-based regression models were estimated: the Generalized Linear Model (GLM), the Generalized Linear Mixed Model (GLMM), and a Distributional Gamma regression model that allows covariates to influence both the mean and dispersion parameters of the outcome distribution. All models assumed that maternal age at first birth follows a Gamma distribution, which is appropriate for modeling positive, continuous, and right-skewed variables.

Each model was estimated using both classical maximum likelihood estimation (MLE) and Bayesian inference, enabling comparison of model fit and predictive performance across estimation frameworks. Model comparison criteria for the classical models included the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and the log-likelihood, while the Bayesian models were evaluated using the Widely Applicable Information Criterion (WAIC), leave-one-out cross-validation (LOO), and the Deviance Information Criterion (DIC).

Classical model comparison

Under the classical estimation framework, the distributional Gamma regression model demonstrated the best overall fit among the competing models (Table 3). This model achieved the lowest AIC (−72,375.6) and lowest BIC (−71,245.3) values, indicating superior performance when balancing model fit and complexity. The model also produced the highest log-likelihood value, suggesting improved agreement with the observed data relative to the standard GLM and GLMM specifications.

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Table 3. Comparative Performance of Gamma-Based Models for Maternal Age at First Birth under Classical and Bayesian Frameworks.

https://doi.org/10.1371/journal.pone.0346504.t003

The improved performance of the distributional Gamma model likely reflects its ability to model both the conditional mean and the dispersion of maternal age at first birth simultaneously, thereby capturing heterogeneity in the variability of the outcome across different socio-demographic groups.

Bayesian model comparison

Under the Bayesian estimation framework, model comparison based on WAIC, LOO cross-validation, and DIC produced consistent results. The Bayesian distributional Gamma regression model achieved the lowest values for all three criteria (WAIC = −1870, LOO = −1876, and DIC = −1865), indicating the best predictive performance among the candidate models. These results suggest that allowing covariates to influence both the location (mean) and dispersion components of the Gamma distribution improves the model’s ability to capture variability in maternal age at first birth among women who had already experienced first birth.

Taken together, the results from both classical and Bayesian comparisons indicate that the distributional Gamma regression model provides the best overall fit to the data. This model was therefore selected as the primary framework for interpreting the associations between maternal age at first birth and the explanatory variables considered in this study.

Although the distributional Gamma regression model demonstrated the strongest overall performance, estimating multiple model frameworks (GLM, GLMM, and distributional regression) provides additional insight into the robustness of the results. Comparing these models allows assessment of how different modeling assumptions such as accounting for unobserved heterogeneity or modeling dispersion affect the estimated associations between predictors and maternal age at first birth. This comparative approach helps evaluate the consistency of coefficient estimates, the stability of effect directions, and the predictive adequacy of the models, thereby strengthening confidence in the empirical findings while acknowledging the limitations inherent in observational cross-sectional data.

Classical Gamma regression model

We recognize that excluding women who had not yet given birth, especially younger and potentially more socioeconomically advantaged individuals, introduces a form of selection bias. For example, if highly educated urban women tend to postpone childbearing beyond the survey period, their systematic omission could lead to underestimation of the association between urban residence, higher education, and delayed first birth. This truncation of the outcome distribution may therefore bias parameter estimates toward earlier ages. Our results should thus be interpreted as describing the determinants of age at first birth among women who have already had a child, rather than the determinants of timing for all women of reproductive age.

Table 4 presents the estimation results of both the Generalized Linear Model (GLM) and Generalized Linear Mixed Model (GLMM) using a gamma distribution with a log link. Both models produced similar coefficient estimates, reflecting consistency in the direction and magnitude of effects.

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Table 4. Parameter Estimates of Maternal Age at First Birth from Gamma Regression Models (GLM and GLMM) with Geographic, Residential and Religious Covariates.

https://doi.org/10.1371/journal.pone.0346504.t004

After adjusting for region and religion, rural women who had already given birth had an expected age at first birth 4.5% lower than that of their urban counterparts (p < 0.05). Controlling for region and residence, mothers who had previously given birth and identified as Orthodox or Protestant exhibited significantly higher average maternal ages at first birth by 4% and 3%, respectively compared to Muslim mothers (the reference group), with results statistically significant at p < 0.05. In contrast, mothers in category others (Catholic, traditional, and other) religions had an expected maternal age at first birth that was 7.5% lower than that of Muslim mothers.

There are clear association in between different regions and maternal age at first birth (AFB) relative to Tigray. Women who had already had child in Amhara (β = –0.047, p < 0.001), Oromia (β = –0.032, p = 0.025), Benishangul (β = –0.032, p = 0.033), and Gambela (β = –0.084, p < 0.001) experienced significantly earlier first births, with Gambela showing the strongest negative effect. By contrast, women in Somali (β = 0.038, p = 0.018), Addis Ababa (β = 0.10, p < 0.001), and Dire Dawa (β = 0.032, p = 0.015) initiated childbearing at older ages, with Addis Ababa exhibiting the largest positive effect, suggesting substantially delayed AFB. Differences in Afar (β = –0.046, p = 0.977) and Harari (β = 0.016, p = 0.297) were not statistically significant, while SNNPR (β = –0.028, p = 0.053) showed borderline evidence of earlier first births compared to Tigray. These numerical patterns underscore the presence of strong geographic disparities in the timing of first birth in Ethiopia.

Sensitivity analysis

To assess the robustness of the Gamma regression results and the potential impact of omitted socioeconomic variables, we re-estimated the model including maternal education, marital status, occupation, age at first marriage, and health service access alongside the original geographic, religious, and residential covariates.

As shown in Table 5, the inclusion of these additional predictors resulted in only marginal changes to the coefficients for region, religion, and residence, with all percentage changes in magnitude remaining below 10%. The statistical significance and direction of these effects were preserved. For example, the coefficient for rural residence changed from –0.04589 to –0.04112 (10.4%), while the coefficient for Addis Ababa decreased slightly from 0.10594 to 0.09987 (5.7%). Similarly, the coefficients for Orthodox and Protestant religious affiliation decreased by 3.5% and 5.0%, respectively, but remained statistically significant.

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Table 5. Sensitivity Analysis: Parameter Estimates of Maternal Age at First Birth from GLM Gamma Regression Models with and Without Additional Socioeconomic Variables.

https://doi.org/10.1371/journal.pone.0346504.t005

In the expanded model, several of the added socioeconomic covariates were themselves significant predictors of maternal age at first birth. Higher maternal education (secondary or above), later age at first marriage, marital status, occupation, and better health service access were all positively associated with older ages at first birth.

The best-performing specification identified in the model comparison (Table 3) was the distributional Gamma regression model, which simultaneously models the mean (location) and dispersion components of the outcome. The mean sub-model represents the expected maternal age at first birth, while the dispersion sub-model captures variability around that mean. This modeling framework is particularly useful because maternal age at first birth among women who had already experienced a first birth may vary not only in its expected value but also in the degree of variability across socio-demographic groups.

As shown in Table 6, the intercept in the mean sub-model is 2.929 (p < 0.001). Under the log link function, exponentiation this coefficient indicates that the reference group defined as urban Muslim women residing in Tigray has an expected maternal age at first birth of approximately 18.7 years. In the dispersion sub-model, the intercept is 3.534 (p < 0.001), representing the baseline level of variability in maternal age at first birth for the same reference group. These intercepts provide the baseline against which the effects of the explanatory variables on both the expected age and its variability can be evaluated.

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Table 6. Classical Gamma Regression Estimates of Mean and Dispersion Parameters for Maternal Age at First Birth by Region, Religion, and Residence in Ethiopia.

https://doi.org/10.1371/journal.pone.0346504.t006

After adjusting for religion and place of residence, notable regional differences in maternal age at first birth remain evident. Relative to Tigray, mothers who had already experienced a first birth in Addis Ababa, Somali, and Dire Dawa had significantly higher expected maternal ages at first birth by approximately 11.2%, 4.2%, and 4.0%, respectively (p < 0.01). These estimates indicate comparatively later initiation of childbearing among women in these regions within the observed sample.

Conversely, women who had gave birth in Amhara, Oromia, and Gambela exhibited significantly lower expected maternal ages at first birth by approximately 4.4%, 3.0%, and 7.8%, respectively (p < 0.05), indicating earlier childbearing relative to the reference region. The magnitude of the association was largest for Gambela. For SNNPR and Benishangul-Gumuz, the estimated reductions in maternal age at first birth were approximately 2.7%, with statistical significance close to conventional thresholds (p ≈ 0.05), suggesting weaker evidence of earlier childbearing relative to Tigray. In contrast, the estimated effects for Harari and Afar were small and statistically insignificant, indicating no clear difference in expected maternal age at first birth compared with the reference region. Overall, these findings suggest substantial geographic heterogeneity in maternal age at first birth among women who had already experienced a first birth in Ethiopia.

Religion was also associated with variation in maternal age at first birth. Compared with Muslim women (reference category), Orthodox Christian women had a significantly higher expected maternal age at first birth of approximately 4.4% (exp(0.043) − 1) × 100%, p < 0.001. Similarly, Protestant women exhibited an expected maternal age at first birth approximately 3% higher than that of Muslim women.

In contrast, women belonging to other religious groups (including Catholic, traditional beliefs, and other affiliations) had an expected maternal age at first birth approximately 5% lower than that of Muslim women, and this difference was statistically significant. These results indicate that maternal age at first birth varies across religious groups within the sample of women who had already experienced childbirth.

After controlling for region and religion, women who had already experienced a first birth and were residing in rural areas had a significantly lower expected maternal age at first birth compared with their urban counterparts. Specifically, the estimated coefficient (β = −0.046) corresponds to an approximately 4.5% lower expected age at first birth for rural women relative to urban women [1 − exp(−0.046)]×100% = 4.5%,p < 0.001[1 − \exp(−0.046)] \times 100\% = 4.5\%, p < 0.001[1 − exp(−0.046)]×100% = 4.5%,p < 0.001. This result indicates that, within the study sample of women who had already given birth, rural residence is associated with comparatively earlier initiation of childbearing than urban residence.

In a Gamma regression, the shape parameter is the inverse of the dispersion (variability). A higher shape implies lower dispersion (more homogeneous maternal age at first birth), and a lower shape implies higher dispersion (greater heterogeneity). Negative coefficients thus imply greater variability in maternal age at first birth compared to the reference category.

The dispersion sub-model examines factors associated with variability in maternal age at first birth among women who had already experienced a first birth. In this parameterization, negative coefficients correspond to lower shape parameters, which indicate greater dispersion (higher variability) in the age at first birth relative to the reference category.

Across regions, most estimated coefficients are negative and statistically significant, suggesting that maternal age at first birth is more variable in many regions compared with the reference region, Tigray. Women who had already experienced a first birth in Afar (β = −0.26) exhibited a substantially lower shape parameter, indicating higher variability in maternal age at first birth. Similarly, Benishangul-Gumuz (β = −0.32) and Harari (β = −0.22) show relatively large negative coefficients, suggesting greater heterogeneity in the timing of first birth among women in these regions.

Moderate but statistically significant increases in variability were also observed in Amhara (β = −0.18), Oromia (β = −0.20), and SNNPR (β = −0.24), indicating wider dispersion in maternal age at first birth relative to Tigray. In contrast, Addis Ababa (β = 0.24) and Dire Dawa (β = 0.14) exhibited positive coefficients in the dispersion model, corresponding to higher shape parameters and therefore lower variability in maternal age at first birth compared with the reference region. This suggests that the timing of first birth among women in these regions is relatively more consistent within the observed sample.

Regarding religion, Orthodox Christian women exhibited significantly greater variability in maternal age at first birth compared with Muslim women (reference category) (β = −0.150, p = 0.008). This negative coefficient indicates a lower shape parameter and therefore greater dispersion in the age at first birth within this group. In contrast, the dispersion estimates for Protestant women and those belonging to other religious groups were not statistically different from the Muslim reference category, suggesting similar levels of variability in maternal age at first birth.

After accounting for regional and religious differences, rural women who had already experienced a first birth exhibited approximately 7% greater variability in maternal age at first birth compared with urban women. This result indicates that the timing of first birth among rural women in the sample is more dispersed than among their urban counterparts.

Overall, the dispersion model highlights notable differences in the consistency of maternal age at first birth across regions, religions, and place of residence, suggesting that variability in the timing of first childbirth differs across socio-demographic contexts within Ethiopia.

Bayesian Gamma regression model

To support the classical estimation, Bayesian inference using MCMC was conducted for GLM GLMM and distributional gamma regression. Proper Bayesian diagnostics are required to ensure convergence, model reliability, and posterior accuracy (Gelman, 2013). The potential scale reduction factor (R) was employed to assess convergence across chains. As shown in Table 7, all R values were below the threshold of 1.01, confirming convergence for all model parameters.

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Table 7. Posterior Summaries and Convergence Diagnostics from Bayesian Gamma Regression Models (GLM and GLMM) for Maternal Age at First Birth.

https://doi.org/10.1371/journal.pone.0346504.t007

Visual inspection of trace plots (Fig 5) for parameters such as the intercept and Afar region further supports this conclusion. Each chain displayed overlapping paths with no systematic trends, indicative of successful exploration of the posterior distribution.

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Fig 5. Post-warmup trace plot for the slope on intercept and (region) 2 (Afar).

https://doi.org/10.1371/journal.pone.0346504.g005

Another alternative way to assess the fit of the Bayesian models, posterior predictive check was conducted. This approach involves generating replicated datasets from the posterior predictive distribution and comparing these simulations to the observed data [43]. The posterior predictive check (PPC) of the mean age at first birth (AFB) demonstrates that the model’s simulated datasets closely reproduce the distribution observed in the empirical data. The black curve, in Fig 6 representing the observed mean AFB distribution, peaks around 17–18 years and exhibits a gradual decline toward higher ages. The green curves, representing the posterior predictive simulations, overlap almost entirely with the observed distribution, indicating that the model accurately captures the central tendency, shape, and spread of the data. This strong alignment suggests that the Bayesian model provides an adequate fit to the observed mean AFB values, with no substantial bias or distortion in its predictions.

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Fig 6. Graphical posterior predictive check for posterior predictive distribution of AFB.

https://doi.org/10.1371/journal.pone.0346504.g006

The agreement between the observed and simulated curves in Fig 6 across the entire distribution, including both tails, highlights the model’s capacity to reflect the variability present in the data. The close correspondence implies that the prior specification, likelihood choice, and posterior estimates together provide a realistic generative mechanism for the observed patterns of AFB. Consequently, this PPC offers strong evidence of model adequacy in representing the key distributional characteristics of mean AFB, reinforcing the validity of subsequent inferences and policy-relevant conclusions drawn from the model outputs.

The other numerical critical diagnostic is the effective sample size (ESS) or , which reflects the number of independent samples equivalent in information content to the correlated MCMC draws. Table 7 shows that ESS values exceed 1000 for all estimated parameters in models, satisfying standard convergence and precision criteria for Bayesian analysis [43].

The Monte Carlo standard error (MCSE) provides insight into the accuracy of MCMC approximations. A low MCSE relative to the posterior standard deviation indicates reliable posterior estimation. In Table 7 both GLM and GLMM models, all parameter MCSE values were minimal and well below the benchmark of 0.1, affirming that the posterior estimates are precise and the sampling process efficient [44].

The (Gelman–Rubin diagnostic) values reported in Table 7 provide evidence of convergence for the Bayesian Gamma regression models under both GLM and GLMM frameworks. A value close to 1.0 indicates that the Markov Chain Monte Carlo (MCMC) chains have mixed well and converged to the target posterior distribution. In these results, all covariates show R values ranging from 0.999 to 1.002, which is well within the commonly accepted threshold of <1.01, confirming excellent convergence across parameters. Even the shape parameter in the GLMM model, with R = 1.01, remains within acceptable limits, suggesting only a minimal deviation but still indicative of reliable convergence. Overall, these diagnostics demonstrate that the Bayesian estimation process was stable and produced trustworthy posterior summaries for both GLM and GLMM models.

As shown in Table 3, the distributional Gamma regression model estimated under the Bayesian framework demonstrated better predictive performance compared with the standard Gamma Generalized Linear Model (GLM) and the Gamma Generalized Linear Mixed Model (GLMM). This model jointly estimates both the mean (location) and the dispersion (shape) components of maternal age at first birth (AFB). The mean sub-model, specified with a log-link function, estimates the expected age at first birth as a function of the explanatory variables, while the dispersion sub-model captures differences in variability across regions, religions, and residential settings (Table 8).

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Table 8. Bayesian Gamma Regression Estimates of Mean and Dispersion Parameters for Maternal Age at First Birth by Region, Religion, and Residence in Ethiopia.

https://doi.org/10.1371/journal.pone.0346504.t008

Convergence diagnostics indicate that the Bayesian estimation performed well. All R values are equal to 1.00, indicating excellent convergence of the Markov Chain Monte Carlo (MCMC) simulations. In addition, the Bulk Effective Sample Size (Bulk_ESS) and Tail Effective Sample Size (Tail_ESS) values are sufficiently large, suggesting stable and reliable posterior estimates for all parameters.

The log-link function ensures that predicted values for both the mean and the shape parameters remain positive. Bayesian estimation was implemented using the No-U-Turn Sampler (NUTS) algorithm, which provides robust posterior sampling and credible interval estimation. The intercept estimate for the mean sub-model (β = 2.93) represents the expected log maternal age at first birth for the reference group defined as women residing in Tigray, belonging to the Muslim religion, and living in urban areas. After exponentiation, this corresponds to an expected maternal age at first birth of approximately 18.7 years (exp(2.93)), with a 95% credible interval of [18.2, 19.3] years. In the dispersion sub-model, the intercept estimate (β = 3.53) represents the baseline shape parameter, indicating relatively low variability in maternal age at first birth for the reference group.

The results presented in Table 8 indicate notable regional variation in maternal age at first birth among women who had already experienced a first birth, after adjusting for religion and place of residence. Relative to Tigray, women residing in Addis Ababa had an estimated 11.6% higher expected age at first birth (95% credible interval: 7.3%–15%), while those in Dire Dawa had approximately 4% higher expected age (95% credible interval: 1%–7.3%). These positive associations indicate comparatively later initiation of childbearing among women in these regions within the observed sample.

In contrast, women in several regions exhibited significantly lower expected ages at first birth relative to Tigray. For example, women residing in Gambela had an estimated 7.7% lower expected age at first birth (95% credible interval: 5%–10%). Similarly, reductions in expected age at first birth were observed in Amhara (4%), Oromia (3%), SNNPR (2.9%), and Benishangul-Gumuz (3%), with credible intervals that did not include zero, indicating statistically meaningful differences relative to the reference region. Overall, these estimates suggest substantial geographic heterogeneity in maternal age at first birth among women who had already experienced childbirth.

Religion was also associated with differences in the expected age at first birth. After adjusting for region and place of residence, women identifying as Orthodox Christian had an expected maternal age at first birth approximately 4% higher than that of Muslim women (β = 0.04; 95% credible interval: 0.03–0.06). Similarly, Protestant women had an expected age at first birth approximately 3% higher than that of Muslim women (β = 0.03; 95% credible interval: 0.02–0.05).

In contrast, women categorized under other religions (including Catholic, traditional beliefs, and other affiliations) had an expected maternal age at first birth approximately 5% lower than that of Muslim women (β = −0.05; 95% credible interval: −0.07 to −0.02). These results indicate differences in maternal age at first birth across religious groups within the study sample.

After adjusting for region and religion, rural residence was associated with a lower expected maternal age at first birth compared with urban residence. Women who had already experienced a first birth and resided in rural areas had an expected age at first birth approximately 5% lower than that of urban women (β = −0.05; 95% credible interval: −0.06 to −0.03). This indicates comparatively earlier initiation of childbearing among rural women within the sample.

In the case of precision (shape) or inverse dispersion sub-model region, religion and place of residence were associated with the maternal age at first birth variability. Positive coefficients for each covariate categories means, higher precision and lower dispersion (variability).

A tight way to interpret shape sub-model is changing shape results in to change in dispersion by the formula dispersion = (exp(−)−1)×100%. After adjusting residence and religion, most regions except Addis Ababa and Dere-Dewa show negative coefficients in the shape sub-model. Women who had already give birth in Somali have about 36.3% (] 100%) higher dispersion (variability), while those in Benishangul exhibit a 37.7% higher variability. Similarly, SNNPR (27.1%), Gambela (32.3%), and Harari (25.9%) also display greater heterogeneity in maternal age at first birth. This suggests that in these regions, women’s age at first birth is much more diverse, with some starting childbearing very early and others delaying, reflecting socio-cultural and economic differences. In contrast, Addis Ababa and Dire Dawa show positive coefficients for the shape parameter, meaning they are associated with lower variability.

Specifically, Addis Ababa has about a 22.1% reduction in dispersion, while Dire Dawa shows a 13.1% reduction relative to the Tigray. This indicates that in urban and administrative centers, women’s age at first birth tends to cluster more tightly around the mean, with fewer extremes of very early or very late first births. Such homogeneity could reflect greater access to education, reproductive health services, family planning and delay of early marriage. Amhara, Oromia, and Afar also show elevated variability, though to a more moderate degree. For instance, Amhara has a 19.7% higher dispersion; Oromia has a 22.1% higher dispersion, and Afar a 29.8% higher dispersion. These moderate increases suggest that, although the majority of women still enter childbearing at relatively young ages, there is growing variation, possibly reflecting changing norms among educated or urban subgroups compared to traditional practices in rural and pastoralist areas.

Religious affiliation also predicts variability. Relative to the Muslim reference group (assuming your coding), Orthodox Christians show a 16.2% higher dispersion and those identifying as other religions have a 18.5% higher dispersion, suggesting greater heterogeneity in the timing of first birth. Protestants, however, show a smaller and uncertain effect, with only an 8.3% increase in dispersion, and the 95% CrI crosses zero, indicating no strong evidence of a difference. These results align with prior demographic studies that find religious affiliation shapes fertility norms and the age of family formation differently across Ethiopian communities.

Rural women exhibit 7.3% greater dispersion in the age at first birth compared with urban women. This means that rural areas not only tend to have earlier mean ages at first birth but also a wider spread, reflecting more inequality in reproductive trajectories. While many rural women begin childbearing very early, a subset may delay slightly due to educational opportunities, regional migration, or access to health services, thereby widening the variability compared to more uniform urban experiences.

The primary rationale for employing three modeling frameworks GLM, GLMM, and distributional Gamma regression was to assess whether the associations between key covariates and maternal age at first birth remained consistent across statistical models of increasing flexibility. The results reported in Tables 4, 6–8 show that both Maximum Likelihood Estimation (MLE) and Bayesian estimation produced coefficient estimates that were broadly consistent in both magnitude and direction across the three modeling approaches. This agreement suggests that the observed relationships between region, religion, place of residence, and maternal age at first birth are relatively stable and not strongly dependent on the specific estimation framework used. Such consistency across estimation methods provides additional confidence in the robustness of the findings.

At the same time, moving from simpler models such as the GLM to more flexible approaches including the GLMM and distributional Gamma regression provided methodological advantages. The GLM offers a useful baseline for examining associations between predictors and the mean maternal age at first birth. The GLMM extends this framework by accounting for potential clustering and unobserved heterogeneity through random effects, which may improve estimation when observations are not fully independent. The distributional Gamma regression model further extends the analysis by allowing covariates to influence not only the mean of maternal age at first birth but also its dispersion. This approach provides a more comprehensive representation of the data structure by capturing both differences in expected age and differences in variability across socio-demographic groups. Consequently, although the substantive conclusions regarding the direction of covariate effects remain similar across models, the more flexible models provide additional insight into heterogeneity in the timing of first birth.

Taken together, the results from the distributional Gamma regression model provide a more detailed understanding of maternal age at first birth among women who had already experienced a first birth in Ethiopia. By jointly modeling the mean and dispersion of the outcome, this framework highlights not only which groups tend to have relatively earlier or later first births, but also where the timing of first birth is more or less variable across subpopulations.

The mean sub-model indicates that maternal age at first birth differs across regions, religious groups, and place of residence. In particular, women residing in urban areas and those living in regions such as Addis Ababa and Dire Dawa tend to exhibit higher expected ages at first birth compared with the reference region. Conversely, lower expected ages at first birth were observed in several regions including Gambela, Amhara, Oromia, SNNPR, and Benishangul-Gumuz. These results highlight geographic and socio-demographic differences in the timing of first childbirth within the study population.

The dispersion (shape) sub-model further reveals that variability in maternal age at first birth also differs across regions and residential settings. Several regions including Afar, Benishangul-Gumuz, Harari, Gambela, Somali, SNNPR, Oromia, and Amhara exhibit lower shape parameters relative to Tigray, indicating greater dispersion in the age at first birth among women in these areas. In contrast, regions such as Addis Ababa and Dire Dawa show relatively higher shape parameters, corresponding to lower variability in the timing of first birth within the observed sample.

Differences in variability were also observed across religious groups. While Orthodox Christian women showed higher expected maternal age at first birth in the mean model, the dispersion model indicates greater variability in the timing of first birth within this group relative to Muslim women. In contrast, Protestant women and those belonging to other religious groups showed no statistically meaningful differences in variability compared with the reference category.

Place of residence also showed consistent associations with both the expected age and variability of first birth. Rural women who had already experienced a first birth exhibited lower expected ages at first birth compared with urban women, and the dispersion model indicates greater variability in maternal age at first birth among rural residents. These results suggest that both the timing and the consistency of first birth differ between rural and urban populations within the sample.

Overall, the distributional Gamma regression framework provides a comprehensive perspective by simultaneously examining differences in both the average timing and the variability of maternal age at first birth. This dual modeling approach helps characterize heterogeneity in reproductive timing across regions, religions, and residential settings among Ethiopian women who had already experienced childbirth.

Strength and limitation of the study

The strengths include reliance on nationwide data and the employment of advanced statistical techniques in both mean and dispersion of maternal age at first birth, as well as adequate models which enhance the confidential applicability and reliability of the results. Moreover, it is believed that the Demographic and Health Survey (DHS), a widely accepted and validated instrument with a large, carefully curated sample size, further strengthens the credibility and accuracy of the results of this study.

This study has several limitations the authors need to acknowledge. First, it focused only on women who had given birth, excluding censored data, which may bias results and limit generalizability. Second, reliance on self-reported DHS data may introduce recall or reporting errors, especially for age at first birth and reproductive behaviors. Third, the cross-sectional design precludes causal inference between covariates and maternal age at first birth. Fourth, unmeasured confounders including maternal education, occupation, early marriage, marital status, household income, healthcare access and quality, cultural practices, and reproductive health knowledge may influence results. Fifth, regional and rural-urban classifications may mask intra-region heterogeneity, particularly in mobile or pastoralist populations, and age reporting errors may affect estimates of mean and variability. Finally, while the distributional Gamma regression captures mean and dispersion effectively, its parametric assumptions may not fully reflect extreme values or nonlinear patterns. Longitudinal studies incorporating broader socio-economic and cultural factors are warranted to strengthen causal inference and inform policy.